General Betting
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28 Mar 12
15:53
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Date Joined:
28 May 11
You have £1,000.
You have an opportunity to play a game betting heads or tails on 10,000 separate coin tosses.
The odds when you bet are evens on both heads and tails (as if the coin was fair).
The coin may be biased.
You are given the information that the chance of heads on a toss is within 40% and 60% with a confidence interval of 99.8%. Assume a normal distribution with mean 50%.
The probability of heads or tails is fixed, but your estimate should fluctuate based on the number of tosses you have observed.
How do you play this game?
The start is easy: on the first toss, you don't bet, as you have no edge. If the first toss is heads, you know heads is slightly more likely on the next toss, as the information you have now makes a bias to heads more likely, so you should bet, but how much?
How do you go through the entire sequence of 10,000 bets. What's the most efficient algorithm?
You have an opportunity to play a game betting heads or tails on 10,000 separate coin tosses.
The odds when you bet are evens on both heads and tails (as if the coin was fair).
The coin may be biased.
You are given the information that the chance of heads on a toss is within 40% and 60% with a confidence interval of 99.8%. Assume a normal distribution with mean 50%.
The probability of heads or tails is fixed, but your estimate should fluctuate based on the number of tosses you have observed.
How do you play this game?
The start is easy: on the first toss, you don't bet, as you have no edge. If the first toss is heads, you know heads is slightly more likely on the next toss, as the information you have now makes a bias to heads more likely, so you should bet, but how much?
How do you go through the entire sequence of 10,000 bets. What's the most efficient algorithm?
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Replies: 9
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Working with evidential probabilities will allow you to test the assumptions you make regarding any edge you have against what's actually happened.
One step up from a guesstimate  |
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First thought would be Bayesian updating to estimate Probability(heads) with the output being used to estimate edge, then using that for kelly.
But that wouldn't use this information: "You are given the information that the chance of heads on a toss is within 40% and 60% with a confidence interval of 99.8%." I think you would could work how to incorporate that into your bayesian prior, but I'd have to think about it. |
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Bayesian would be the way to go. Usually for bernoulli trials you'd use a beta distribution for the conjugate prior.
If we know however that it is normally distributed it's harder. The fact it's within 40-60 with a 99.8% gives us a variance but as with the above poster I'm not sure how you'd use it to seed your prior. It is probably something like picking the right intial beta parameters based on variance. |
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Perhaps it might be more useful to most of us " average" punters on here if you just cut to the chase Invester, and clearly and straightforwardly outline the point you are trying to make here ?
And also then tell us equally clearly how it can improve our betting strategies, if that is in fact the case ?. Or do we really have to be able to solve these puzzles by ourselves inorder to gain full benefit from your knowledge ? |
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Btw I hate even doing crosswords so you can guess where I'm coming from.
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Simply use a binomial confidence interval calculator.
http://statpages.org/confint.html No bets until both High and Low are in the same direction away from 0.5. |
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Simply ?
For some maybe. |
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It is a problem that may be solved by using The Central Limit Theorem ,using random variables and deviation etc.It is beyond my limited capabilities.
If anyone is interested in the Kelly System there is a free software package,a small one,available from The University Of Amsterdam via http://staff.feweb.vu.nl/tijms There are several interesting items on it. |
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If you enjoy puzzles I have posted a few on my blog for your delectation over the Easter period.
They do require an understanding of the fundamentals of probability theory but they are not too difficult. If I get time I will post some easier ones too. |